快速小波转换（英语：Fast wavelet transform）是利用数学的演算法则用来转换在时域的波形或信号变成一系列的以正交基底（英语：Orthogonal basis）构成的小而有限的波、小波。 当然，快速小波转换本身可以很轻易地扩增它的维度以符合各种不同的需求，例如影像处理、压缩、去除噪声…等 ${\displaystyle S_{n}^{(J)}:=2^{J}\left\langle {f(t),\phi (2^{J}t-n)}\right\rangle }$

利用${\displaystyle S^{(M)))$且M<J的一系列常数集合，以及由${\displaystyle d^{(K)))$，k=M,1,...,J-1的差分集合 可以导出有递回关系式的数学式如下： ${\displaystyle S_{n}^{(k+1)}:=\sum \limits _{k=-N}^{N}{a_{k}S_{2n-k}^{(k)}+\sum \limits _{k=-N}^{N}{b_{k}d_{2n-k}^{(k)))))$

或是导入Z转换，以k=J-1,J-2,...,M且${\displaystyle n\in {\mathbb {Z} ))$可改写为

${\displaystyle S^{(k+1)}(Z)=a(Z)\cdot (\uparrow 2)(s^{(k)}(Z))+b(Z)\cdot (\uparrow 2)(d^{(k)}(Z))}$

其中${\displaystyle (\uparrow 2)}$表示升高采样操作子

G. Beylkin, R. Coifman, V. Rokhlin, "Fast wavelet transforms and numerical algorithms" Comm. Pure Appl. Math., 44 (1991) pp. 141–183

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